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Properties

Label 14.4.162...367.1
Degree $14$
Signature $[4, 5]$
Discriminant $-1.624\times 10^{20}$
Root discriminant \(27.77\)
Ramified primes $3,13,109$
Class number $1$
Class group trivial
Galois group $C_2^4.\PSL(2,7)$ (as 14T42)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^14 - 6*x^13 + 9*x^12 + 25*x^11 - 80*x^10 + 104*x^9 - 425*x^8 + 1261*x^7 + 5*x^6 - 4607*x^5 + 6299*x^4 + 754*x^3 - 5512*x^2 + 702*x + 1453)
 
Copy content gp:K = bnfinit(y^14 - 6*y^13 + 9*y^12 + 25*y^11 - 80*y^10 + 104*y^9 - 425*y^8 + 1261*y^7 + 5*y^6 - 4607*y^5 + 6299*y^4 + 754*y^3 - 5512*y^2 + 702*y + 1453, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 6*x^13 + 9*x^12 + 25*x^11 - 80*x^10 + 104*x^9 - 425*x^8 + 1261*x^7 + 5*x^6 - 4607*x^5 + 6299*x^4 + 754*x^3 - 5512*x^2 + 702*x + 1453);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^14 - 6*x^13 + 9*x^12 + 25*x^11 - 80*x^10 + 104*x^9 - 425*x^8 + 1261*x^7 + 5*x^6 - 4607*x^5 + 6299*x^4 + 754*x^3 - 5512*x^2 + 702*x + 1453)
 

\( x^{14} - 6 x^{13} + 9 x^{12} + 25 x^{11} - 80 x^{10} + 104 x^{9} - 425 x^{8} + 1261 x^{7} + 5 x^{6} + \cdots + 1453 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $14$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[4, 5]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(-162420703254885611367\) \(\medspace = -\,3^{7}\cdot 13^{6}\cdot 109^{5}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(27.77\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $3^{1/2}13^{3/4}109^{1/2}\approx 123.80306325772969$
Ramified primes:   \(3\), \(13\), \(109\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-327}) \)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11\cdots 49}a^{13}+\frac{40\cdots 78}{11\cdots 49}a^{12}+\frac{45\cdots 53}{11\cdots 49}a^{11}+\frac{14\cdots 48}{11\cdots 49}a^{10}+\frac{84\cdots 99}{11\cdots 49}a^{9}+\frac{30\cdots 69}{11\cdots 49}a^{8}-\frac{13\cdots 51}{11\cdots 49}a^{7}+\frac{11\cdots 22}{11\cdots 49}a^{6}-\frac{36\cdots 35}{11\cdots 49}a^{5}+\frac{20\cdots 57}{11\cdots 49}a^{4}-\frac{22\cdots 68}{11\cdots 49}a^{3}-\frac{34\cdots 74}{11\cdots 49}a^{2}-\frac{58\cdots 41}{11\cdots 49}a-\frac{12\cdots 36}{11\cdots 49}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{2}$, which has order $2$
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $8$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{42\cdots 23}{11\cdots 49}a^{13}-\frac{22\cdots 19}{11\cdots 49}a^{12}+\frac{26\cdots 77}{11\cdots 49}a^{11}+\frac{11\cdots 19}{11\cdots 49}a^{10}-\frac{26\cdots 24}{11\cdots 49}a^{9}+\frac{32\cdots 77}{11\cdots 49}a^{8}-\frac{17\cdots 75}{11\cdots 49}a^{7}+\frac{45\cdots 20}{11\cdots 49}a^{6}+\frac{18\cdots 84}{11\cdots 49}a^{5}-\frac{16\cdots 95}{11\cdots 49}a^{4}+\frac{17\cdots 02}{11\cdots 49}a^{3}+\frac{65\cdots 56}{11\cdots 49}a^{2}-\frac{10\cdots 16}{11\cdots 49}a-\frac{34\cdots 09}{11\cdots 49}$, $\frac{22\cdots 83}{91\cdots 73}a^{13}-\frac{99\cdots 46}{91\cdots 73}a^{12}+\frac{79\cdots 13}{91\cdots 73}a^{11}+\frac{81\cdots 90}{91\cdots 73}a^{10}-\frac{82\cdots 31}{91\cdots 73}a^{9}+\frac{48\cdots 89}{91\cdots 73}a^{8}-\frac{68\cdots 20}{91\cdots 73}a^{7}+\frac{13\cdots 00}{91\cdots 73}a^{6}+\frac{40\cdots 58}{91\cdots 73}a^{5}-\frac{88\cdots 48}{91\cdots 73}a^{4}-\frac{74\cdots 77}{91\cdots 73}a^{3}+\frac{16\cdots 77}{91\cdots 73}a^{2}-\frac{45\cdots 00}{91\cdots 73}a-\frac{78\cdots 32}{91\cdots 73}$, $\frac{69\cdots 30}{11\cdots 49}a^{13}-\frac{61\cdots 85}{11\cdots 49}a^{12}+\frac{17\cdots 61}{11\cdots 49}a^{11}+\frac{59\cdots 11}{11\cdots 49}a^{10}-\frac{97\cdots 23}{11\cdots 49}a^{9}+\frac{20\cdots 78}{11\cdots 49}a^{8}-\frac{51\cdots 42}{11\cdots 49}a^{7}+\frac{17\cdots 52}{11\cdots 49}a^{6}-\frac{23\cdots 95}{11\cdots 49}a^{5}-\frac{28\cdots 92}{11\cdots 49}a^{4}+\frac{11\cdots 19}{11\cdots 49}a^{3}-\frac{10\cdots 56}{11\cdots 49}a^{2}-\frac{11\cdots 78}{11\cdots 49}a+\frac{50\cdots 41}{11\cdots 49}$, $\frac{64\cdots 81}{11\cdots 49}a^{13}-\frac{33\cdots 88}{11\cdots 49}a^{12}+\frac{32\cdots 53}{11\cdots 49}a^{11}+\frac{17\cdots 85}{11\cdots 49}a^{10}-\frac{35\cdots 78}{11\cdots 49}a^{9}+\frac{43\cdots 51}{11\cdots 49}a^{8}-\frac{25\cdots 78}{11\cdots 49}a^{7}+\frac{62\cdots 48}{11\cdots 49}a^{6}+\frac{43\cdots 84}{11\cdots 49}a^{5}-\frac{24\cdots 31}{11\cdots 49}a^{4}+\frac{21\cdots 33}{11\cdots 49}a^{3}+\frac{14\cdots 13}{11\cdots 49}a^{2}-\frac{14\cdots 42}{11\cdots 49}a-\frac{60\cdots 35}{11\cdots 49}$, $\frac{24\cdots 15}{11\cdots 49}a^{13}-\frac{12\cdots 84}{11\cdots 49}a^{12}+\frac{93\cdots 05}{11\cdots 49}a^{11}+\frac{71\cdots 27}{11\cdots 49}a^{10}-\frac{12\cdots 31}{11\cdots 49}a^{9}+\frac{12\cdots 81}{11\cdots 49}a^{8}-\frac{90\cdots 13}{11\cdots 49}a^{7}+\frac{21\cdots 39}{11\cdots 49}a^{6}+\frac{22\cdots 19}{11\cdots 49}a^{5}-\frac{89\cdots 40}{11\cdots 49}a^{4}+\frac{60\cdots 75}{11\cdots 49}a^{3}+\frac{84\cdots 42}{11\cdots 49}a^{2}-\frac{50\cdots 54}{11\cdots 49}a-\frac{35\cdots 24}{11\cdots 49}$, $\frac{42\cdots 89}{11\cdots 49}a^{13}-\frac{22\cdots 23}{11\cdots 49}a^{12}+\frac{24\cdots 33}{11\cdots 49}a^{11}+\frac{10\cdots 39}{11\cdots 49}a^{10}-\frac{20\cdots 47}{11\cdots 49}a^{9}+\frac{28\cdots 77}{11\cdots 49}a^{8}-\frac{18\cdots 96}{11\cdots 49}a^{7}+\frac{45\cdots 56}{11\cdots 49}a^{6}+\frac{11\cdots 56}{11\cdots 49}a^{5}-\frac{11\cdots 82}{11\cdots 49}a^{4}+\frac{80\cdots 97}{11\cdots 49}a^{3}+\frac{12\cdots 20}{11\cdots 49}a^{2}-\frac{75\cdots 02}{11\cdots 49}a-\frac{46\cdots 95}{11\cdots 49}$, $\frac{23\cdots 14}{11\cdots 49}a^{13}-\frac{12\cdots 91}{11\cdots 49}a^{12}+\frac{10\cdots 43}{11\cdots 49}a^{11}+\frac{68\cdots 92}{11\cdots 49}a^{10}-\frac{12\cdots 98}{11\cdots 49}a^{9}+\frac{13\cdots 14}{11\cdots 49}a^{8}-\frac{89\cdots 41}{11\cdots 49}a^{7}+\frac{21\cdots 48}{11\cdots 49}a^{6}+\frac{20\cdots 49}{11\cdots 49}a^{5}-\frac{90\cdots 94}{11\cdots 49}a^{4}+\frac{65\cdots 76}{11\cdots 49}a^{3}+\frac{78\cdots 46}{11\cdots 49}a^{2}-\frac{56\cdots 20}{11\cdots 49}a-\frac{36\cdots 60}{11\cdots 49}$, $\frac{54\cdots 88}{11\cdots 49}a^{13}-\frac{43\cdots 85}{11\cdots 49}a^{12}+\frac{12\cdots 83}{11\cdots 49}a^{11}+\frac{27\cdots 00}{11\cdots 49}a^{10}-\frac{68\cdots 08}{11\cdots 49}a^{9}+\frac{12\cdots 57}{11\cdots 49}a^{8}-\frac{31\cdots 40}{11\cdots 49}a^{7}+\frac{13\cdots 58}{11\cdots 49}a^{6}-\frac{14\cdots 36}{11\cdots 49}a^{5}-\frac{22\cdots 22}{11\cdots 49}a^{4}+\frac{68\cdots 71}{11\cdots 49}a^{3}-\frac{40\cdots 74}{11\cdots 49}a^{2}-\frac{21\cdots 07}{11\cdots 49}a+\frac{18\cdots 21}{11\cdots 49}$ Copy content Toggle raw display
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 49614.4197132 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{5}\cdot 49614.4197132 \cdot 1}{2\cdot\sqrt{162420703254885611367}}\cr\approx \mathstrut & 0.304983645817 \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^14 - 6*x^13 + 9*x^12 + 25*x^11 - 80*x^10 + 104*x^9 - 425*x^8 + 1261*x^7 + 5*x^6 - 4607*x^5 + 6299*x^4 + 754*x^3 - 5512*x^2 + 702*x + 1453) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^14 - 6*x^13 + 9*x^12 + 25*x^11 - 80*x^10 + 104*x^9 - 425*x^8 + 1261*x^7 + 5*x^6 - 4607*x^5 + 6299*x^4 + 754*x^3 - 5512*x^2 + 702*x + 1453, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 6*x^13 + 9*x^12 + 25*x^11 - 80*x^10 + 104*x^9 - 425*x^8 + 1261*x^7 + 5*x^6 - 4607*x^5 + 6299*x^4 + 754*x^3 - 5512*x^2 + 702*x + 1453); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 6*x^13 + 9*x^12 + 25*x^11 - 80*x^10 + 104*x^9 - 425*x^8 + 1261*x^7 + 5*x^6 - 4607*x^5 + 6299*x^4 + 754*x^3 - 5512*x^2 + 702*x + 1453); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2^4.\PSL(2,7)$ (as 14T42):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 2688
The 22 conjugacy class representatives for $C_2^4.\PSL(2,7)$
Character table for $C_2^4.\PSL(2,7)$

Intermediate fields

7.3.2007889.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 14 sibling: data not computed
Degree 28 siblings: data not computed
Degree 42 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.7.0.1}{7} }^{2}$ R ${\href{/padicField/5.14.0.1}{14} }$ ${\href{/padicField/7.3.0.1}{3} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ R ${\href{/padicField/17.3.0.1}{3} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.7.0.1}{7} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.7.0.1}{7} }^{2}$ ${\href{/padicField/47.7.0.1}{7} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.7.2.7a1.2$x^{14} + 4 x^{9} + 2 x^{7} + 4 x^{4} + 4 x^{2} + 4$$2$$7$$7$$C_{14}$$$[\ ]_{2}^{7}$$
\(13\) Copy content Toggle raw display $\Q_{13}$$x + 11$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{13}$$x + 11$$1$$1$$0$Trivial$$[\ ]$$
13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.2.4.6a1.1$x^{8} + 48 x^{7} + 872 x^{6} + 7200 x^{5} + 24216 x^{4} + 14400 x^{3} + 3488 x^{2} + 397 x + 16$$4$$2$$6$$C_8$$$[\ ]_{4}^{2}$$
\(109\) Copy content Toggle raw display $\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
109.2.1.0a1.1$x^{2} + 108 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
109.1.2.1a1.2$x^{2} + 654$$2$$1$$1$$C_2$$$[\ ]_{2}$$
109.2.2.2a1.2$x^{4} + 216 x^{3} + 11676 x^{2} + 1296 x + 145$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
109.2.2.2a1.2$x^{4} + 216 x^{3} + 11676 x^{2} + 1296 x + 145$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)